where \(\mathbb {w}_{k}\)  is the Wigner component in Eqs. (REF )-() (corresponding to: \(\mathbb {w}_{0} = \mathbb {s}\)  , \(\mathbb {w}_{1} = \mathbb {v}_{0}\)  , \(\mathbb {w}_{2} = \mathbb {v}_{1}\)  and \(\mathbb {w}_{3} = \mathbb {p}\) ), and \(\mathbb {w}_{vac}\)  is the corresponding vacuum initial condition Eq. (REF ).
Total energy of the Dirac particle is
\(\varepsilon \left( x , p_{x} , t \right) = {m \mathbb {s}^{v} \left( x , p_{x} , t \right) + p_{x} \mathbb {v}_{1}^{v} \left( x , p_{x} , t \right)}\) ,
and the particle number density in phase space can be written as [1]}
\(n \left( x , p_{x} , t \right) = \frac{m \mathbb {s}^{v} \left( x , p_{x} , t \right) + p_{x} \mathbb {v}_{1}^{v} \left( x , p_{x} , t \right)}{\Omega \left( p_{x} \right)},\) 
